Harmonic Electric Field Applied to a Particle in an Infinite Square Well
This Demonstration shows the evolution of the wavefunction for a particle as it makes a transition between eigenstates of an infinite square well. Transitions are caused by a harmonic electric field applied along the width of the well, where the driving frequency is assumed to be equal to the natural frequency separation between the initial and final states (quantum numbers and , respectively), to within a "detuning frequency" . The dimensionless electric field magnitude is . Once an initial state is selected, only those final states for which the electric dipole matrix element is nonzero are shown. The time-dependent Schrödinger equation is solved by an expansion in square-well eigenstates. The initial and final states are included automatically, and additional eigenstates (with quantum number ) can be added.[more]
The top figure shows (red curve) along with the direction and relative strength of the electric field (blue arrow). Also shown is the potential energy function (thick black line). The expectation value of the particle energy and the normalization of are shown at the top of the plot. All quantities are dimensionless, with , where is the well width, and and are the particle's mass and charge, respectively.
The bottom figure shows the probability of being found in the initial or final states as a function of time (solid black and solid blue curves, respectively). Also shown is the probability of being in the final state, as calculated from first-order perturbation theory (dashed blue curve).[less]
Consider a particle of mass and charge in a 1D infinite potential well between and . Suppose we perturb this system by applying an electric field , so that the potential energy inside the well becomes . In terms of the characteristic energy of the well , the electric field magnitude can be written as , for some dimensionless parameter .
The particle is assumed to be initially in an eigenstate of the unperturbed well, with quantum number . If we want to cause a transition to another unperturbed eigenstate, with quantum number , the chance of this is maximized by having the driving frequency (the electric field frequency in this case) equal to the natural frequency separation between the initial and final eigenstates (which is equal to the energy difference divided by ). Of course, this fine tuning may not be possible, and the "detuning frequency" is the difference between the optimal driving frequency and the actual one.
The time-dependent Schrödinger equation is solved by expanding the wavefunction in terms of the unperturbed stationary states of the well
where is the energy of level . At a minimum, the expansion should include the initial and final states. However, additional states can be added to explore their effect (which is fairly minimal!). Since is a separable function of space and time, the spatial integrations need only be performed once, and the matrix elements of are precalculated for the first 10 energy levels.
We also show the time-dependent probability of finding the particle in the initial or final states, along with the transition probability calculated from first-order perturbation theory. From perturbation theory, the probability of being found in the final state at time is given by
where is the dipole matrix element.
Snapshots show the to transition for .
Snapshot 1: initial time, when the particle is in the state
Snapshot 2: intermediate time, when the particle is about equally likely to be found in the or states upon measurement
Snapshot 3: at time , where the probability is close to unity for being found in the state